Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $x \neq 0$. $p = \dfrac{-10}{10(3x - 4)} \div \dfrac{-9}{3x - 4} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{-10}{10(3x - 4)} \times \dfrac{3x - 4}{-9} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ -10 \times (3x - 4) } { 10(3x - 4) \times -9 } $ $ p = \dfrac {-10 (3x - 4)} {-9 \times 10(3x - 4)} $ $ p = \dfrac{-10(3x - 4)}{-90(3x - 4)} $ We can cancel the $3x - 4$ so long as $3x - 4 \neq 0$ Therefore $x \neq \dfrac{4}{3}$ $p = \dfrac{-10 \cancel{(3x - 4})}{-90 \cancel{(3x - 4)}} = -\dfrac{10}{-90} = \dfrac{1}{9} $